~ (3, CHAPTER P Preparation for Calculus. Section P.1 Graphs and Models. 1. y = --~x+3 x-intercept: (2, 0) y-intercept: (0, 3) Matches graph (b).

Transcription

1 CHAPTER P Preparation for Calculus Section P.1 Graphs and Models 1. = --~x+3 x-intercept: (2, 0) -intercept: (0, 3) Matches graph (b). 2. =x/-~-x 2 x-intercepts: (-3, 0), (3, O) -intercept: (0, 3) Matches graph (d). 3. =3-x 2 x-intercepts: (W/-~, 0), (-~f~, O) -intercept: (0, 3) Matches graph (a). 4. =X3--X x-intercepts: (0, 0), (-1, 0), (1, 0) -intercept: (0, 0) Matches graph (c). 5. =½X+2 X (o, 2) 7. =4-x 2 x /-2" (0, 4) 8. = (x- 3) 2 ~ (3, X i(o, 9) 1(6,9) 8, ~(1, " ~(4,11)~ 4) /5, 4) 2- \(2, 1)/ x 9. =Ix+2 x (-4, O) -2- ~(-5, 3) 4-6. =5-2x X i (-3,11)V ~ -6-4 (-2, O) I I / " (3;0) -2- -(- 1, 1) I 2 4- (o, ~ 5), 3) -4 2

2 Section P. 1 Graphs and Models x+2 x x _1 _1 _1 4-1 Undef. 1,~,,,~2) 2-1)- (- 1, o)/-~ -2 ~, x(0, - 1) 2) (-1, t)~,[ = x/rfx - 6 X Xmin = -5 Xmax = 4 Xscl = 1 min = -5 max = 8 scl = 1 Note that = -3 when x = 0 and = 0 when -4" -6" 0, -6) -8" 12. =~x+2 x Xmin = -20 Xmax = 30 Xscl = 5 min = -10 max = 40 scl = 5 Note that = 16 when x = 0 or =.,/~ - x 3" (-4.00, 3) (o,~) (-2, o) =-- x 5 I x _3-3 Undef (a) (2, ) = (2, 1.73) ( = ~-2 = ~ ~ 1.73) (b) (x, 3) = (-4, 3) (3 : 45 - (-4)) 18. = x s -5x 3 ~ ~(1, 3) -9 ~(I, -4) -3/ -2~ -6 (a) (-0.5, ) = (-0.5, 2.47) (b) (x,-4) = (-1.65,-4)and(x,-4) = (1,-4),o

3 4 Chapter P Preparation for Calculus 19. = 2x - 5 -intercept: = 2(0)- 5 = -5; (0,-5) x-intercept: 0 = 2x- 5 5 = 2x 20. = 4x intercept: = 4(0) = 3, (0, 3) x-intercept: 0 = 4x = 4x 2 None, cannot equal = x 2 +x-2 -intercept: = = -2; (0,-2) x-intercepts: 0 = x 2 + x = X 3 _ 4x 0 = (x+2)(x-1) -intercept: 2 = 0 3 _ 4(0) = o; (o, o) x = -2, 1; (-2, 0), (1, O) x-intercepts: 0 = x 3-4x 23. = x-,jl6- x 2 0 = x(x-2)(x+2) x = 0, +2; (0, 0), (+-2, 0) -intercept: = Ox/ri~ = O; (0, O) x-intercepts: 0 = x-,/]6 - x = (x-1)~x = x4(4 - x)(4 + x) x ~ O, 4,-4; (0, 0), (4, 0), (-4, O) -intercept: = (0-1)-,J-~ + 1 = -1; (0,-1) x-intercept: 0 = (x -1)~x x = 1; (1, o) x -intercept: None. x cannot equal 0. x-intercept: 0-5x 0=2-~x x 2 + 3x (3x + 1) 2 -intercept: - x = 4; (4, O) (0) I-3(o) + = o; (o, o) x 2 + 3x x-intercepts: 0 - (3x + 1) X(X + 3) (3x + 1) x2 -- X = 0 x = O,-3; (0, 0), (-3, O) -intercept: 02() = 0 = 0; (0, 0) x-intercept: x2(o)- x 2 + 4(0) = 0 x = o; (o, o) 28. = 2x-~x2 +1 -intercept: = 2(0) -,fo = -1; (0,-1) x-intercept: 0 = 2x- xi~x ~ + 1 2x = ~xa +1 4X 2 = X x 2 = 1 X X t. ~]~ 3 x = --~-;,0 Note: x = -~/3 is an extraneous solution. 29. Smmetric with respect to the -axis because = -x -6 = x 2-6.

4 Section P. 1 Graphs and Models 30, = x z - x No smmetr with respect to either axis or the origin. 31. Smmetric with respect to the x-axis because (_)2 = 2 = X 3 _ 8X, 41. = 2-3x Intercepts: (0, 2), (-}, 0) Smmetr: None 32, Smmetric with respect to the origin because (--) = (--X) 3 + (--X) - = -x 3 - x =x3+x. (o, 2) I 33, Smmetric with respect to the origin because (-x)(-) = x = Smmetric with respect to the x-axis because x(-)~ : x~ =-~o. 35. = 4-~x+3 No smmetr with respect to either axis or the origin. 36. Smmetric with respect to the origin because (-x)(-) - N/4 -(-x) 2 = 0 x-,,/7- x 2 = O. 37, Smmetric with respect to the origin because --X -(_x)~+~ x -x2+l" x = x2 +---~ is smmetric with respect to the -axis --X) _ X 2 because - (-x) x = Ix 3 + X[ is smmetric with respect to the -axis e.us = (-x/ +I = I-(x + x/i = + xl. 40. ] - x = 3 is smmetric with respect to the x-axis because ]-x = =--}x+6 Intercepts: (0, 6), (4, 0) Smmetr: None ~(0, 6) 2" 43. = ½x - 4 (4,0) I ~x Intercepts: (8, 0), (0,-4) Smmetr: none = 32-x lo 8 10 O, -4) Intercepts: (0, 1), (--}, O) Smmetr: none 2 (o, 1) I I ~-x 1 2

5 6 Chapter P Preparation for Calculus = 9 - x 2 Intercepts: (0, 9), (3, 0), (-3, O) Smmetr: -axis 49. = x Intercepts: (_3~, 0), (0, 2) Smmett?: none 4-3" _+_21_ 46. = x Intercept: (0, 3) Smmetr: -axis 50. = x 3-4x Intercepts: (0, 0), (2, 0), (-2, O) Smmetr: origin -6-3 I = (x + 3) 2 Intercepts: (-3, 0), (0, 9) Smmetr: none (0, 9) 51. = x,,/~x + 5 Intercepts: (0, 0), (-5, O) Smmetr: none 3 2 (-5, o) (o, o) (-3,0)_ = 2x 2 +x = x(2x+l) Intercepts: (0, 0), (-½, O) Smmetr none 52. = -,/25 - x 2 Intercepts: (0, 5), (5, 0), (-5, O) Smmetr: -axis (o, 5) x

6 Section P. 1 Graphs and Models 53. x = 3 Intercept: (0, 0) Smmetr: origin 57. = 6-Ix[ Intercepts: (0, 6), (-6, 0), (6, 0) Smmetr: -axis 3-2- (o, o) ~4" 54. x = 2 _ 4 Intercepts: (0, 2), (0,-2), (-4, 0) Smmetr: x-axis 3 (o, 2) - x " =[6-xI Intercepts: (0, 6), (6, 0) Smmetr: none ~, 6) I ~--x (0, -2) =- Intercepts: none Smmetr: origin _ x = 9 2 = x+9 = +_-,f~- + 9 Intercepts: (0, 3), (0,-3), (-9, O) Smmetr: x-axis 4 I I I I =~5+l Intercept: (0, 10) Smmetr: -axis (o, ~o) 60. x = 4 ~ = Intercepts: (-2, 0), (2, 0), (0,-1), (0, 1) Smmetr: origin and both axes Domain: [-2, 2] I t( I

7 Chapter P Preparation for Calculus 61. x = 6 3 ~- = 6 - x =+ ~ Intercepts: (6, 0), (0, x/~), (0,-~/~) Smmetr: x-axis,(o, 4~) ~. ----~.(o,-4~) 62. 3x-4 2 = = 3x - 8 = +N~-]x - 2 0) Smmetr: x-axis 63. x+ = 8 ~ = 8-x 4x- = 7 ~ = 4x-7 8-x=4x-7 15 = 5x 3=X The corresponding -value is = 5. Point of intersection: (3, 5) 3x x-2 =-4 ~ = x x+ 2 =-10 ~ = 2 3x + 4-4x x+4 =-4x-10 7x = -14 x = -2 The corresponding -value is = -1. Point of intersection: (-2, -1) 65. x 2 + = 6 ~ = 6- x 2 x+=4~ =4-x 6-x 2 = 4-x 0=xZ-x-2 0 = (x-z)(x+l) x = 2,-1 The corresponding -values are = 2 (for x = 2) and = 5(forx =-1). Points of intersection: (2, 2), (-1, 5) x = 3-2 ~ 2 = 3-x =x-1 3-x=(x-1) 2 3-x = x 2-2x+1 0 = x :-x-2 = (x+l)(x-2) x=-lorx= 2 The corresponding -values are = -2 (for x = -1) and =l(forx = 2). Points of intersection: (-1,-2), (2, 1) 67. x 2 + : = 5 ~ 2 = 5- X 2 x-=l~ =x x~ = (x - 1)2 5-x z = x 2-2x+l 0 = 2x 2-2x-4 = 2(x+l)(x-2) x =-lorx=2 The corresponding -values are = -2 (for x = -1) and = 1 (forx = 2). Points of intersection: (-1,-2), (2, 1) 68. x 2 +2 = 25 ~ ~ = 25-x ~ -3x+ = 15 ~ = 3x x ~ = (3x+15) 2 25-x ~ = 9x 2 +90x = 10x ~ +90x = x 2 + 9x = (x+ 5)(x+4) x =-4or x =-5 The corresponding -values are = 3 (for x = -4) and = 0 (for x = -5). Points of intersection: (-4, 3), (-5, 0)

8 Section P.1 Graphs and Models =X 3 X 3 = X X3--x=O x(x+l)(x-1) : 0 X = 0, X = -1, or x = 1 The corresponding -values are = 0 (forx = 0), =-1 (forx =-l),and = l(for x = 1). Points of intersection: (0, 0), (-1,-1), (1, 1) 71. Analticall, = x 3-4x =-(x + 2) x 3-4x =-(x+2) x 3-3x + 2 = 0 (x-1)2(x+2) = 0 x=lorx=-2 The corresponding -values are =-3 (for x = 1) and = 0 (forx =-2). Points of intersection: (1,-3), (-2, 0) = x 3-2x2 +x-1 = -x 2 + 3x - 1 x 3-2x2 +x-1 =-x ~ +3x-1 x 3 - x 2-2x = 0 x(x-2)(x+l) = 0 x =-1,0,2. Points of intersection: (-1,-5), (0,-1), (2, 1) _/.. 41 c01-i),~,.7~(2,1) i l I 6 - =-x 72. Analticall, = x 4-2X = 1-X a 1-X 2 = X 4-2x X 4 --X 2 o = x2(x + 1)(x- l) x = -1, O, 1. Points of intersection: (-1, 0), (0, 1), (1, O) 73. = ~/7+6 = ~/-x 2-4x 4-2,,, Points of intersection: (-2, 2), (-3, w/-~) ~ (-3, 1.732) Analticall, 74. = -]2x =6-x 7 ",, (1, 5)- ~(3, 3)~ -~7 + 6 = ~-x 2-4x x + 6 = -x 2-4x x 2 +5x+6 = 0 (x+3)(x+2) = 0 x = -3,-2. - [N ---I x-31+6 J Points of intersection: (3, 3), (1, 5) Analticall, -I 2x = 6- x 2x-3 = xor 2x-3 =-x x=3or 12x- 31= x x=l.